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Unitary and Anti-Unitary Quantum Descriptions of the Classical Not Gate

Gianpiero Cattaneo, G. Conte, Roberto Leporini

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Two possible quantum descriptions of the classical Not gate are investigated in the framework of the Hilbert space C2: the unitary and the anti-unitary operator realizations. The two cases are distinguished interpreting the unitary Not as a quantum realization of the classical gate which on a xed orthogonal pair of unit vectors, realizing once for all the classical bits 0 and 1, produces the required transformations 0 => 1 and 1 => 0 (i.e., logical quantum Not). The anti-unitary Not is a quantum realization of a gate which acts as a classical Not on any pair of mutually orthogonal vectors, each of which is a potential realization of the classical bits (i.e., universal quantum Not). Although the latter is not completely positive, one can give an approximated unitary realization of the gate by appending an ancilla. Finally, we consider the unitary and the anti-unitary operator realizations of two important genuine quantum gates that transform elements of the computational basis of C2 into its superpositions: the square root of the identity and the square root of the Not.

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Cambridge University Press eBooks•Feb 19, 2009•0 citations
Two possible quantum descriptions of the classical Not gate are investigated in the framework of the Hilbert space C2: the unitary and the anti-unitary operator realizations. The two cases are distinguished interpreting the unitary Not as a quantum realization of the classical gate which on a xed orthogonal pair of unit vectors, realizing once for all the classical bits 0 and 1, produces the required transformations 0 => 1 and 1 => 0 (i.e., logical quantum Not). The anti-unitary Not is a quantum realization of a gate which acts as a classical Not on any pair of mutually orthogonal vectors, each of which is a potential realization of the classical bits (i.e., universal quantum Not). Although the latter is not completely positive, one can give an approximated unitary realization of the gate by appending an ancilla. Finally, we consider the unitary and the anti-unitary operator realizations of two important genuine quantum gates that transform elements of the computational
basis of C2 into its superpositions: the square root of the identity and the square root of the Not.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.
Quantum gates are fundamental in Quantum computing for their role in manipulating elementary information carriers referred to as quantum bits. In this paper, a theoretical scheme for realizing a quantum Hadamard and a quantum controlled-NOT logic gates operations in the anti-Jaynes-Cummings interaction process is provided. Standard Hadamard operation for a specified initial atomic state is achieved by setting a specific sum frequency and photon number in the normalized anti-Jaynes-Cummings qubit state transition operation with the interaction component of the anti-Jaynes-Cummings Hamiltonian generating the state transitions. The quantum controlled-NOT logic gate is realized when a single atomic qubit defined in a two-dimensional Hilbert space is the control qubit and two non-degenerate and orthogonal polarized cavities defined in a two-dimensional Hilbert space make the target qubit. With precise choice of interaction time in the anti-Jaynes-Cummings qubit state transition operations defined in the anti-Jaynes-Cummings sub-space spanned by normalized but non-orthogonal basic qubit state vectors, ideal unit probabilities of success in the quantum controlled-NOT operations is determined.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.